Uniqueness Theorem Complex Analysis

Question

I am having some trouble solving the following question for my practical final.

Does there exist an entire function f : C → C s.t. $f(1/n) = \frac{n^2}{n^2 + 1 }$ for all n in N

I don't think there exists an entire function. I know we have to use the uniqueness principle. My attempt to prove this is by contradiction:

So I know that $f(0)$ = $\lim\limits_{x \to inf} \ f(1/n)$ = 1

Now let the set E be the {0} U { 1/n} which has a limit point and let $g(z) = \frac{1/z^2}{1/z^2 + 1 }$ which has a singularity at - i and + i. I am not sure how to proceed. Thank you!

Answer 1

As you noted yourself, $g(z) = \dfrac {1/z^2}{1/z^2+1} = \dfrac {1}{z^2+1}$ is already a meromorphic function which satisfies your condition, and it has poles at $\pm i$. If there were an entire function $f$ which also satisfied your condition, then it would agree with $g$ on $\{\frac 1n + 0i | n \in \mathbb{N}\} \cup \{0 \}$. Thus it would have to agree with $g$ on a set with a limit point in $\mathbb{C}$, so by analytic continuation, $f$ must be $g$ itself; thus it cannot be entire.

Answer 2

Let $g(z)=(1+z^{2})f(z)-1$. Then $g(\frac 1 n)=0$ for all $n$ so $g(z)=0$ for all $z$. Hence $f(z)=\frac 1 {1+z^{2}}$ which has poles at $\pm i$. Hence you cannot have an entire function satisfying the given property.